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Geometry - Symmetry: Line of Symmetry and Reflection

Grade 6ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A figure is said to be symmetrical if it can be folded along a line such that the two parts coincide exactly. This line is known as the Line of Symmetry or the Axis of Symmetry. For example, an isosceles triangle has one line of symmetry passing through its vertex perpendicular to the base.

An isosceles triangle with a vertical line of symmetry passing through the top vertex.
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A Regular Polygon (where all sides and angles are equal) has as many lines of symmetry as the number of its sides. For instance, a square has 4 lines of symmetry, while a regular pentagon has 5.

A square showing four lines of symmetry: vertical, horizontal, and two diagonals.
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Reflection Symmetry (or Mirror Symmetry) occurs when an object is reflected across a mirror line. The image formed is at the same perpendicular distance from the mirror line as the object, but on the opposite side. Note that reflection causes Lateral Inversion, meaning the left side of the object appears as the right side of the image.

An L-shaped object reflected across a vertical mirror line showing lateral inversion.
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In a coordinate plane, reflecting a point across the xx-axis changes the sign of the yy-coordinate, while reflecting across the yy-axis changes the sign of the xx-coordinate. The line of reflection acts as the perpendicular bisector of the segment joining the object and the image.

📐Formulae

Number of lines of symmetry in a regular polygon=n\text{Number of lines of symmetry in a regular polygon} = n

Perpendicular distance of object from mirror line=Perpendicular distance of image from mirror line\text{Perpendicular distance of object from mirror line} = \text{Perpendicular distance of image from mirror line}

Total distance between object and its image=2×distance from mirror line\text{Total distance between object and its image} = 2 \times \text{distance from mirror line}

Reflection of point (x,y) across the x-axis→(x,−y)\text{Reflection of point } (x, y) \text{ across the x-axis} \rightarrow (x, -y)

Reflection of point (x,y) across the y-axis→(−x,y)\text{Reflection of point } (x, y) \text{ across the y-axis} \rightarrow (-x, y)

💡Examples

Problem 1:

Identify the number of lines of symmetry for a rectangle and a circle.

Solution:

  1. For a rectangle: It has 22 lines of symmetry. These are the lines joining the midpoints of the opposite sides (one vertical and one horizontal). Note that the diagonals of a rectangle are NOT lines of symmetry.
  2. For a circle: It has an infinite number of lines of symmetry. Any line passing through the center of the circle (the diameter) acts as a line of symmetry.

Explanation:

A line of symmetry must divide the shape into two identical halves that coincide when folded. In a rectangle, folding along a diagonal does not align the corners, so only the lines connecting midpoints work. A circle is perfectly round, so any diameter splits it into two equal semicircles.

Problem 2:

A point AA is located 7 cm7 \text{ cm} away from a mirror line LL. If A′A' is the reflected image of AA, calculate the total distance between the original point AA and its image A′A'.

Solution:

Step 1: Note the distance of the object from the mirror line: d=7 cmd = 7 \text{ cm}. Step 2: Use the property of reflection that the image A′A' is at the same distance behind the mirror line as the object is in front of it. Distance of A′A' from L=7 cmL = 7 \text{ cm}. Step 3: Calculate total distance AA′=Distance of A to L+Distance of A′ to LAA' = \text{Distance of } A \text{ to } L + \text{Distance of } A' \text{ to } L. AA′=7 cm+7 cm=14 cmAA' = 7 \text{ cm} + 7 \text{ cm} = 14 \text{ cm}.

Explanation:

The mirror line is the perpendicular bisector of the line segment joining the object and its image. Therefore, the total distance is simply double the distance from the object to the mirror.

Problem 3:

Draw the lines of symmetry for an Equilateral Triangle and state the total number of lines of symmetry.

An equilateral triangle with three lines of symmetry intersecting at the center.

Solution:

An equilateral triangle has 3 lines of symmetry. Each line passes through a vertex and the midpoint of the opposite side. Total number of lines of symmetry = 33.

Explanation:

Since an equilateral triangle is a regular polygon with 3 equal sides, it follows the rule that a regular polygon of nn sides has nn lines of symmetry.

Problem 4:

A point P(4,3)P(4, 3) is reflected in the yy-axis to form image P′P'. Find the coordinates of P′P' and the distance between PP and the yy-axis.

Coordinate plane showing point P at (4,3) and its reflection P' at (-4,3) across the y-axis.

Solution:

  1. Reflection in the yy-axis changes the sign of the xx-coordinate: P(4,3)→P′(−4,3)P(4, 3) \rightarrow P'(-4, 3).
  2. The distance of point P(4,3)P(4, 3) from the yy-axis is the absolute value of its xx-coordinate: ∣4∣=4|4| = 4 units.

Explanation:

The yy-axis acts as the mirror line. The perpendicular distance from (4,3)(4, 3) to the line x=0x=0 (the yy-axis) is 44 units. The image must be 44 units on the other side, at x=−4x = -4.